Examples

Chapter 13 Class 11 Limits and Derivatives
Serial order wise

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Example 20 Find the derivative of f(x) from the first principle, where f(x) is (ii) x sin x Given f (x) = x sin x We need to find Derivative of f(x) We know that f’(x) = lim┬(h→0) 𝑓⁡〖(𝑥 + ℎ) − 𝑓(𝑥)〗/ℎ Here, f (x) = x sin x So, f (x + h) = (x + h) sin (x + h) Putting values f’(x) =lim┬(h→0) ((𝑥 + ℎ) sin⁡〖 (𝑥 + ℎ) − 𝑥 sin⁡〖𝑥 〗 〗)/ℎ Using sin (A + B) = sin A cos B + cos A sin B = lim┬(h→0)⁡〖((𝑥 + ℎ)(sin⁡〖𝑥 cos⁡〖ℎ +〖 cos 𝑥〗⁡sin⁡〖ℎ 〗 )〗 − 𝑥 sin⁡𝑥 〗)/ℎ〗 = lim┬(h→0)⁡〖(𝑥(sin⁡〖𝑥 cos⁡〖ℎ +〖 cos〗⁡〖𝑥 sin⁡〖ℎ) + ℎ (sin⁡〖𝑥 cos⁡〖ℎ +〖 cos〗⁡〖𝑥 sin⁡〖ℎ) − 𝑥 sin⁡𝑥 〗 〗 〗 〗 〗 〗 〗 〗)/ℎ〗 = lim┬(h→0)⁡〖(𝑥 sin⁡〖𝑥 cos⁡〖ℎ + 𝑥 cos⁡〖𝑥 sin⁡〖ℎ + ℎ 〖(sin〗⁡〖𝑥 cos⁡〖ℎ + cos⁡〖𝑥 sin⁡〖ℎ) − 𝑥 sin⁡𝑥 〗 〗 〗 〗 〗 〗 〗 〗)/ℎ〗 = lim┬(h→0)⁡〖(𝑥𝑠𝑖𝑛 𝑥 cos⁡〖ℎ − 𝑥 sin⁡〖𝑥 + 𝑥 cos⁡〖𝑥 sin⁡〖ℎ + ℎ(sin⁡〖𝑥 cos⁡〖ℎ + cos⁡〖𝑥 sin⁡ℎ 〗 〗)〗 〗 〗 〗 〗)/ℎ〗 = lim┬(h→0)⁡〖(𝑥𝑠𝑖𝑛 𝑥 〖(cos〗⁡〖ℎ − 1)+ 𝑥 cos⁡〖𝑥 sin⁡〖ℎ + ℎ(sin⁡〖𝑥 cos⁡〖ℎ + cos⁡〖𝑥 sin⁡ℎ 〗 〗)〗 〗 〗 〗)/ℎ〗 = lim┬(h→0)⁡((𝑥 sin⁡〖𝑥 (cos⁡〖ℎ −1)〗 〗)/ℎ+(𝑥 cos⁡〖𝑥 sin⁡ℎ 〗)/ℎ+(ℎ (sin⁡〖𝑥 cos⁡〖ℎ +cos⁡〖𝑥 sin⁡〖ℎ)〗 〗 〗 〗)/ℎ) = lim┬(h→0)⁡〖〖x sin〗⁡〖𝑥 (cos⁡〖ℎ −1)〗 〗/ℎ+lim┬(h→0) (𝑥 cos⁡〖𝑥 sin⁡ℎ 〗 )/h+lim┬(h→0) (sin⁡〖𝑥 cos⁡〖ℎ +cos⁡〖𝑥 sin⁡〖ℎ )〗 〗 〗 〗 〗 = – x sin x (𝐥𝐢𝐦)┬(𝐡→𝟎) ((𝟏−〖 𝒄𝒐𝒔〗⁡〖𝒉)〗)/𝒉+𝑥 cos⁡〖𝑥 (𝐥𝐢𝐦)┬(𝐡→𝟎) 𝒔𝒊𝒏⁡𝒉/𝐡+〗 lim┬(h→0) (sin⁡〖𝑥 cos⁡〖ℎ +cos⁡〖𝑥 sin⁡〖ℎ )〗 〗 〗 〗 = – x sin x (0) + x cos x (1) + ( sin x cos 0 + cos x sin 0) = 0 + x cos x + sin × 1 + cos x × 0 = 0 + x cos x + sin x + 0 = x cos x + sin x Hence f’ (x) = x cos x + sin x